Optimal. Leaf size=124 \[ \frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 \sqrt{c-\frac{c}{a x}}}+\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{4 a \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{4 a^2} \]
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Rubi [A] time = 0.239776, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {6178, 863, 873, 875, 208} \[ \frac{c x^2 \sqrt{1-\frac{1}{a^2 x^2}}}{2 \sqrt{c-\frac{c}{a x}}}+\frac{c x \sqrt{1-\frac{1}{a^2 x^2}}}{4 a \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{4 a^2} \]
Antiderivative was successfully verified.
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Rule 6178
Rule 863
Rule 873
Rule 875
Rule 208
Rubi steps
\begin{align*} \int e^{\coth ^{-1}(a x)} \sqrt{c-\frac{c}{a x}} x \, dx &=-\left (c \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{x^2}{a^2}}}{x^3 \sqrt{c-\frac{c x}{a}}} \, dx,x,\frac{1}{x}\right )\right )\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 \sqrt{c-\frac{c}{a x}}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x^2 \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{4 a}\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{4 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 \sqrt{c-\frac{c}{a x}}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{c-\frac{c x}{a}}}{x \sqrt{1-\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{8 a^2}\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{4 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 \sqrt{c-\frac{c}{a x}}}+\frac{c^2 \operatorname{Subst}\left (\int \frac{1}{-\frac{c}{a^2}+\frac{c^2 x^2}{a^2}} \, dx,x,\frac{\sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{4 a^4}\\ &=\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x}{4 a \sqrt{c-\frac{c}{a x}}}+\frac{c \sqrt{1-\frac{1}{a^2 x^2}} x^2}{2 \sqrt{c-\frac{c}{a x}}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{1-\frac{1}{a^2 x^2}}}{\sqrt{c-\frac{c}{a x}}}\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.324908, size = 148, normalized size = 1.19 \[ \frac{2 a^2 x^2 \sqrt{1-\frac{1}{a^2 x^2}} (2 a x+1) \sqrt{c-\frac{c}{a x}}+\sqrt{c} (1-a x) \log \left (2 a^2 \sqrt{c} x^2 \sqrt{1-\frac{1}{a^2 x^2}} \sqrt{c-\frac{c}{a x}}+c \left (2 a^2 x^2-a x-1\right )\right )+\sqrt{c} (a x-1) \log (1-a x)}{8 a^2 (a x-1)} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.17, size = 102, normalized size = 0.8 \begin{align*} -{\frac{x}{8}\sqrt{{\frac{c \left ( ax-1 \right ) }{ax}}} \left ( -4\,{a}^{3/2}x\sqrt{ \left ( ax+1 \right ) x}-2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{ \left ( ax+1 \right ) x}\sqrt{a}+2\,ax+1 \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{{\frac{ax-1}{ax+1}}}}}{a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ \left ( ax+1 \right ) x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} x}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.20489, size = 674, normalized size = 5.44 \begin{align*} \left [\frac{{\left (a x - 1\right )} \sqrt{c} \log \left (-\frac{8 \, a^{3} c x^{3} - 7 \, a c x - 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}} - c}{a x - 1}\right ) + 4 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{16 \,{\left (a^{3} x - a^{2}\right )}}, \frac{{\left (a x - 1\right )} \sqrt{-c} \arctan \left (\frac{2 \,{\left (a^{2} x^{2} + a x\right )} \sqrt{-c} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{2 \, a^{2} c x^{2} - a c x - c}\right ) + 2 \,{\left (2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + a x\right )} \sqrt{\frac{a x - 1}{a x + 1}} \sqrt{\frac{a c x - c}{a x}}}{8 \,{\left (a^{3} x - a^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{c - \frac{c}{a x}} x}{\sqrt{\frac{a x - 1}{a x + 1}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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